Module Identifier MA31210  
Academic Year 2004/2005  
Co-ordinator Dr Robert J Douglas  
Semester Semester 2  
Course delivery Lecture   19 x 1 hour lectures  
  Seminars / Tutorials   3 x 1 hour example classes  
Assessment TypeAssessment Length/DetailsProportion
Semester Exam2 Hours (written examination)  100%
Supplementary Assessment2 Hours (written examination)  100%

Learning outcomes

On completion of this module, a student should be able to:

Brief description

A wide variety of phenomena can be modelled by means of ordinary differential equations.
Very few such equations can be solved explicitly, and in the quantitative theory of differential equations methods have been developed to determine the behaviour of solutions directly from the equation itself. The subject was pioneered in the early part of the twentieth century by Poincare and then by Liapunov. This module provides a thorough grounding in the theory of dynamical systems and nonlinear differential equations.


To provide an introduction to the qualitative theory of nonlinear differential equations, with particular emphasis on the construction of phase portraits of two-dimensional systems and applications.


1. Existence and uniqueness of solutions; autonomous and non-autonomous systems.
2. One-dimensional systems; stability and invariance of solutions.
3. Two-dimensional linear systems: classification of critical points.
4. Critical points of two-dimensional nonlinear systems. Increased stability, Poincare-Benedixson theorem an its consequences; construction of possible phase portraits.
5. Modelling by means of two-dimensional nonlinear systems, eg predator-prey models, infectious disease models.
6. Singular pertubations and matched asymptotic expansions. Law of mass action in chemical reactions.

Reading Lists

** Supplementary Text
D W Jordan & Smith (1987) Nonlinear ordinary differential equations. 2nd. Oxford University Press 0198596561
M Hisch and S Smale (1974) Differential equations, dynamical systems, and linear algebra. Academic Press
P Glendinning (1994) Stability, instability and chaos : an introduction to the theory of nonlinear differential equations CUP 0-521-42566-2
J Guckenheimer & P Holmes (1983) Nonlinear oscillations, dynamical systems & bifurcations of vector fields Springer 3540908196


This module is at CQFW Level 6